The summability (A) of the conjugate series of a Fourier series

Misra, Mahendra

doi:10.1215/s0012-7094-47-01467-1

Cited by 4 publications

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“…In the year 2002, Misra and Sahoo [20] studied absolute Banach summability of the conjugate series of a Fourier series. Subsequently, Paikray et al [21, 22] studied absolute Banach summability of a factored Fourier series and its conjugate series.…”

Section: Introductionmentioning

confidence: 99%

Approximation and simulation of signals via harmonic Banach summable factors of Fourier series

Sonker

Devi

Jena

et al. 2023

Math Methods in App Sciences

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It is well known that the Fourier series is not summable by harmonic means almost everywhere. This paper aimed to introduce the harmonic Banach product summability factors of Fourier series and conjugate Fourier series in order to establish two new approximation theorems. Moreover, some illustrative examples are considered to obtain the sum of certain oscillatory and non-oscillatory sequences and demonstrated that their convergence are well behaved under the proposed mean. Also, the findings are validated and simulated through MATLAB programming.

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Section: Introductionmentioning

confidence: 99%

Approximation and simulation of signals via harmonic Banach summable factors of Fourier series

Sonker

Devi

Jena

et al. 2023

Math Methods in App Sciences

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“…Many classical results establish the summability of the series from the local behavior of the function [7,8,17,18,32,33,35]; some others go in the opposite direction, they obtain local information about the function itself from certain summability properties of the series [3,10,11,12,13,16,26]. The first problem is of Abelian nature while the second one usually has a Tauberian character [9].…”

Section: Introductionmentioning

confidence: 99%

On the Point Behavior of Fourier Series and Conjugate Series

Estrada¹,

Vindas²

2010

Z. Anal. Anwend.

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We investigate the point behavior of periodic functions and Schwartz distributions when the Fourier series and the conjugate series are both Abel summable at a point. In particular we show that if f is a bounded function and its Fourier series and conjugate series are Abel summable to values γ and β at the point θ_0 , respectively, then the primitive of f is differentiable at θ_0 , with derivative equal to γ , the conjugate function satisfies \lim_{θ→θ_0} \frac{3}{(θ–θ_0)^3} \int^θ_{θ_0} \tilde f(t)(θ–t)^2 dt = β , and the Fourier series and the conjugate series are both (\mathrm{C}, κ) summable at θ_0 , for any κ > 0 . We show a similar result for positive measures and L^1 functions bounded from below. Since the converse of our results are valid, we therefore provide a complete characterization of simultaneous Abel summability of the Fourier and conjugate series in terms of “average point values”, within the classes of positive measures and functions bounded from below. For general L^1 functions, we also give a.e. distributional interpretation of − \frac{1}{2π} p.v. \int^π_{–π} f (t+θ_0) \cot \frac t2 dt as the point value of the conjugate series when viewed as a distribution. We obtain more general results of this kind for arbitrary trigonometric series with coefficients of slow growth, i.e., periodic distributions.

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Notes on Fourier analysis (XXV): Quasi-Tauberian theorem

Sunouchi

1950

Tohoku Math. J. (2)

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The first general treatment of quasi-Tauberian theorems was given by N. Wiener C16J. In this note the author proves another quasi-Tauberian theorem concerning absolute limit under Wiener's conditions. Wiener derived the Cesaro summability theorem of Fourier series frcm his general theorem.In this paper further applications of his theorem and analogues concerning absolute limit are given. Some of theorems proved in this paper are known and the other are new. It is interesting that theεe theorems are derived fi om two key theorems which are given in § 1. In § 2, the Cesaro and absolute Cesaro summability theorems due to Paley [11J and Boεanquet [lj C2j are derived frcm the key theorems. In § 3, we prove the Cesaro and absolute Cesaro summability theorems of the conjugate Fourier series. These theorems include the essential parts of the results due to Paley [llj and Boεanquet-Hyslop [5^. The Cesaro summability problems of the derived Fourier series are discussed in § 4. The essential part of the results due to Takahashi (13^, Wang [14J (Ί53, Zygmund C173, Bosanquet £3^ £4J and Hyslop [TJ are derived frcm our key theorems. It is well known that these results are interpreted as the relation between Cesaro summation and Riemann summation of the first kind and their a nalogues concerning absolute summation. In § 5, the relations between the generalized jump of a function and its Fourier constants are discussed. This problem was treated early by Zygmund [18J and Szεεz Γ12> Theεe results are known as the relation between Cesaro summation, Riemann summation of the second kind and their analogues concerning absolute summation. As applications of these theorems, we can also prove theorems analogous to the results due to Misra C9D and Moursund C10>In the sequel we shall fully use the notations and theorems of Chapter VII in the Wiener's work Clβ^ without references.

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The summability (A) of the conjugate series of a Fourier series

Cited by 4 publications

References 0 publications

Approximation and simulation of signals via harmonic Banach summable factors of Fourier series

Approximation and simulation of signals via harmonic Banach summable factors of Fourier series

On the Point Behavior of Fourier Series and Conjugate Series

Notes on Fourier analysis (XXV): Quasi-Tauberian theorem

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